"The lady doesn't run."
Translation:Ní ritheann an bhantiarna.
The verb is rith, not bí.
All present tense verbs are negated with ní. níl is a special case, where ní and fuil, the present tense depended form of bí (also seen in an bhfuil? and go bhfuil) fuse to form níl.
If the indicative form of the sentence doesn't use tá, the negative form won't use níl.